Aadhar is not meant to be private. It should not be treated a s a private key when you need to give a copy of your card everywhere like your bank, getting a new cell phone sim card, employment, etc. The problem with Aadhar (among other things) is that people think its a secret. Its not supposed to be that way.
SSN is meant to be a secret but it's required at all the places you mentioned. At least with Aadhaar, if someone got a copy of my number, it can't be used without MFA.
